In 2-dimensional Riemann manifold ,Ricci curvature is given by $$ R_{ij}=\frac{1}{2}Rg_{ij} $$ My PDE teachers teach me to compute it by the way. $$ R_{11}=g^{ij}R_{1i1j}=g^{22}R_{1212} \\ R_{12}=g^{21}R_{1221} \\ R_{21}=g^{12}R_{2112} \\ R_{22}=g^{11}R_{2121} \\ R=g^{ij}R_{ij}=2g^{11}g^{22}R_{1212}-2g^{12}g^{12}R_{1212}=2R_{1212} \begin{vmatrix} g^{11} & g^{12} \\ g^{21} & g^{22} \end{vmatrix} \\ \frac{1}{2}Rg_{11}=2R_{1212}g_{11} \begin{vmatrix} g^{11} & g^{12} \\ g^{21} & g^{22} \end{vmatrix} =2R_{1212}g^{22}\begin{vmatrix} g^{11} & g^{12} \\ g^{21} & g^{22} \end{vmatrix}^{-1} \begin{vmatrix} g^{11} & g^{12} \\ g^{21} & g^{22} \end{vmatrix} \\ \frac{1}{2}Rg_{11}=R_{11} $$
Others are likely , but he thinks there should be easy way ,but I can't work out. Whether there are easy way to compute it ?
You can simplify the calculations by working in an orthonormal frame so that contraction with the metric become standard traces. The only non-zero components of the Riemannian curvature are $$K := R_{1212} = R_{2121} = -R_{1221} = -R_{2112}.$$ Then $$R_{ij} = R_{1i1j} + R_{2i2j}= K\delta_{i2}\delta_{j2} + K \delta_{i1}\delta_{j1} = Kg_{ij}.$$
Taking the trace we get $R = 2K$, and thus $$R_{ij} = \frac12 R g_{ij}.$$